tangent spoke

Spoke

A spoke is one of some number of rods radiating from the center of a wheel (the hub where the axle connects), connecting the hub with the round traction surface.

The term originally referred to portions of a log which had been split lengthwise into four or six sections. The radial members of a wagon wheel were made by carving a spoke (from a log) into their finished shape. A spokeshave is a tool originally developed for this purpose. Eventually, the term spoke was more commonly applied to the finished product of the wheelwright's work, than to the materials he used.

Construction

Spokes can be made of wood, metal, or synthetic fiber depending on whether they will be in tension or compression.

Compression spokes

The original type of spoked wheel with wooden spokes was used for horse drawn carriages and wagons. In early motor cars, wooden spoked wheels of the artillery type were normally used.

In a simple wooden wheel, a load on the hub causes the wheel rim to flatten slightly against the ground as the lowermost wooden spoke shortens and compresses. The other wooden spokes show no significant change.

Wooden spokes are mounted radially. They are also dished, usually to the outside of the vehicle, to prevent wobbling. Also, the dishing allows the wheel to compensate for expansion of the spokes due to absorbed moisture by dishing more.

Types

Some types of wheels have removable spokes which can be replaced individually if they break or bend. These include bicycle and wheelchair wheels. High quality bicycles with conventional wheels use spokes of stainless steel, while cheaper bicycles may use galvanized (also called "rustless") or chrome plated spokes. While a good quality spoke is capable of supporting about 225 kgf (c. 500 pounds-force or 2,200 newtons) of tension, they are used at a fraction of this load to avoid suffering fatigue failures. Since bicycle and wheelchair wheel spokes are only in tension, flexible and strong materials such as synthetic fibers, are also occasionally used. Metal spokes can also be ovalized or bladed to reduce aerodynamic drag, and butted (double or even triple) to reduce weight while maintaining strength.

A variation on the wire-spoked wheel was Tioga's "Tension Disk", which appeared superficially to be a solid disk but was in fact constructed using the same principles as a normal tension-spoked wheel. Instead of individual wire spokes, a continuous thread of Kevlar (aramid) was used to lace the hub to the rim under high tension. The threads were encased in a translucent disk for protection and some aerodynamic benefit, but this was not a structural component.

Reaction to load

Pre-tensioned wire-spoked wheels react similarly to a load. The load on the hub causes the wheel rim to flatten slightly against the ground as the lowermost pre-tensioned spoke shortens and compresses, losing some of its pre-tension. Perhaps surprisingly, the upper spokes show no significant change in tension.

For explanations, computer models, and tests confirming this odd behavior, see The Bicycle Wheel by Jobst Brandt, and Figure 10 in http://www.duke.edu/~hpgavin/papers/HPGavin-Wheel-Paper.pdf, which all show the lower spokes of pre-tensioned bicycle wheels losing their pre-tension as they roll under a loaded hub.

Tangential lacing

Wire spokes can be radial to the hub but are more often mounted tangentially to the hub. Tangential spoking allows for the transfer of torque between the rim and the hub. Tangential spokes are thus necessary for the drive wheel, which has torque at the hub from pedalling, and any wheels using disk brakes, which have torque transferred from the rim to the disk (via the hub) when braking.

Wheelbuilding

Constructing a tension-spoked wheel from its constituent parts is called wheelbuilding and requires the correct building procedure for a strong and long-lasting end product. Tensioned spokes are usually attached to the rim or sometimes the hub with a spoke nipple. The other end is commonly peened into a disk or uncommonly bent into a "Z" to keep it from pulling through its hole in the hub. The bent version has the advantage of replacing a broken spoke in a rear bicycle wheel without having to remove the rear gears.

Wire wheels, with their excellent weight to strength ratio, soon became popular for light vehicles. For everyday cars, wire wheels were soon replaced by the less expensive metal disc wheel, but wire wheels remained popular for sports cars up to the 1960s. Spoked wheels are still popular on motorcycles.

Spoke length

When building a bicycle wheel, the spokes must have the right length. If the spokes are too short, they can not be tightened. If they are too long they will touch the rim tape, possibly puncturing the tire.

Calculation

For wheels with crossed spokes (which are the norm), the desired spoke length is

m = number of spokes to be used for one side of the wheel, for example 36/2=18,

k = number of crossings per spoke, for example 3 and

α = 360° k/m.

Regarding a: For a symmetric wheel such as a front wheel with no disc brake, this is half the distance between the flanges. For an asymmetric wheel such as a front wheel with disc brake or a rear wheel with chain derailleur, the value of a is different for the left and right sides.

α is the angle between the radius through the hub hole and the radius through the corresponding spoke hole. The angle between hub hole radii is 360°/m (for evenly spaced holes). For each crossing, one spoke hole further down the hub is used, multiplying the angle by the number of crossings k. For example, a 32 spoke wheel has 16 spokes per side, 360° divided by 16 equals 22.5°. Multiply 22.5° (one cross) by the number of crossings to get the angle - if 3-cross, the 32 spoke wheel has an angle α of 67.5 degrees.

Derivation

The spoke length formula computes the length of the space diagonal of an imaginary rectangular box. Imagine holding a wheel in front of you such that a nipple is at the top. Look at the wheel from along the axis. The spoke through the top hole is now a diagonal of the imaginary box. The box has a depth of a, a height of r2-r1cos(α) and a width of r1sin(α).

Equivalently, the law of cosines may be used to first compute the length of the spoke as projected on the wheel's plane (as illustrated in the diagram), followed by an application of the Pythagorean theorem.